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arXiv:1901.05655 · Physical Review A

Photon blockade and the quantum-to-classical transition in the driven-dissipative Josephson pendulum coupled to a resonator

Iivari Pietikäinen, Jani Tuorila, Dmitry S. Golubev, Gheorghe Sorin Paraoanu

Published Jun 20, 2019• 31 citations
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Abstract

We investigate the driven quantum phase transition between the oscillating motion and the classical nearly free rotations of the Josephson pendulum coupled to a harmonic oscillator in the presence of dissipation. We refer to this as the Josephson-Rabi model. This model describes the standard setup of circuit quantum electrodynamics, where typically a transmon device is embedded in a superconducting cavity. We find that by treating the system quantum mechanically this transition occurs at higher drive powers than expected from an all-classical treatment, which is a consequence of the quasiperiodicity originating in the discrete energy spectrum of the bound states. We calculate the photon number in the resonator and show that its dependence on the drive power is nonlinear. In addition, the resulting multiphoton blockade phenomenon is sensitive to the truncation of the number of states in the transmon, which limits the applicability of the standard Jaynes-Cummings model as an approximation for the pendulum-oscillator system. We calculate the $n\mathrm{th}$-order correlation functions of the blockaded microwave photons and observe the differences between the rotating-wave approximation and the full multilevel Josephson-Rabi Hamiltonian with the counter-rotating terms included. Finally, we compare two different approaches to dissipation, namely the Floquet-Born-Markov and the Lindblad formalisms.

DOI
10.1103/physreva.99.063828

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We investigate the driven quantum phase transition between the oscillating motion and the classical nearly free rotations of the Josephson pendulum coupled to a harmonic oscillator in the presence of dissipation. We refer to this as the Josephson-Rabi model. This model describes the standard setup of circuit quantum electrodynamics, where typically a transmon device is embedded in a superconducting cavity. We find that by treating the system quantum mechanically this transition occurs at higher drive powers than expected from an all-classical treatment, which is a consequence of the quasiperiodicity originating in the discrete energy spectrum of the bound states. We calculate the photon number in the resonator and show that its dependence on the drive power is nonlinear. In addition, the resulting multiphoton blockade phenomenon is sensitive to the truncation of the number of states in the transmon, which limits the applicability of the standard Jaynes-Cummings model as an approximation for the pendulum-oscillator system. We calculate the nthn\mathrm{th}nth-order correlation functions of the blockaded microwave photons and observe the differences between the rotating-wave approximation and the full multilevel Josephson-Rabi Hamiltonian with the counter-rotating terms included. Finally, we compare two different approaches to dissipation, namely the Floquet-Born-Markov and the Lindblad formalisms.
We investigate the driven quantum phase transition between bound oscillating motion and the classical nearly free rotations of the Josephson pendulum coupled to a harmonic oscillator in the presence of dissipation. This model describes the standard setup of circuit quantum electrodynamics, where typically a transmon device is embedded in a superconducting cavity. We find that by treating the system quantum mechanically this transition occurs at higher drive powers than expected from an all-classical treatment, which is a consequence of the quasiperiodicity originating in the discrete energy spectrum of the bound states. We calculate the photon number in the resonator and show that its dependence on the drive power is nonlinear. In addition, the resulting multi-photon blockade phenomenon is sensitive to the truncation of the number of states in the transmon, which limits the applicability of the standard Jaynes-Cummings model as an approximation for the pendulum-oscillator system. We also compare two different approaches to dissipation, namely the Floquet-Born-Markov and the Lindblad formalisms.
The model of a two-level system (TLS) coupled to a harmonic oscillator has found a widespread application in physics and chemistry. In this thesis we focus on the field of circuit quantum electrodynamics where the TLS stands for a quantum bit (qubit), which is the basic building unit of a potential quantum computer, and the oscillator represents a transmission-line resonator, which can be used, e.g., to store information contained on the qubit or for communication between several qubits. Furthermore, the oscillator can model a superconducting quantum interference device (SQUID) determining the state of the qubit. In first experimental realizations the coupling between the qubit and the oscillator was small compared to characteristic frequencies of the two devices. In such a situation the Jaynes-Cummings model provides a realistic and completely analytical description of the energy spectrum and the dynamics of the setup. However, it relies on a rotating-wave approximation (RWA) which neglects counter-rotating terms in the full qubit-oscillator Hamiltonian. These counter-rotating terms become important under increase of the coupling strength, as it has been achieved in recent experiments. We present two analytical approaches to take these additional contributions into account: The first one treats the qubit-oscillator Hamiltonian perturbatively in the coupling strength and predicts a frequency shift in its energy spectrum, the Bloch-Siegert shift. The second one considers the qubit's tunneling splitting as small parameter and thus treats the problem to all orders in the coupling allowing to enter the ultrastrong coupling regime, where the coupling strength becomes comparable to the qubit's and/or oscillator's frequency. For both cases we give a thorough analysis of the qubit's dynamics. In order to perform operations the qubit is usually probed by an external driving force. We model this situation by coupling the TLS to a classical oscillator and examine the resulting time dependent Hamiltonian using a combination of Floquet theory and Van Vleck perturbation theory. Thus, we provide an analysis of the qubit's energy spectrum and dynamics which is nonperturbative in the driving amplitude. We investigate effects like coherent destruction of tunneling and driving-inducing tunneling oscillations. By coupling the driven TLS to a quantized oscillator we give for the first time an analytical treatment of the qubit-oscillator system being simultaneously exposed to both ultrastrong coupling and extreme driving. We observe interesting phenomena in the dynamics like the suppression of specific frequencies under a variation of the coupling strength. To mimic environmental influences on the qubit we apply the Caldeira-Leggett master equation approach to the driven TLS and to the qubit-oscillator system and provide for both cases analytical expressions for the relaxation and dephasing rates.
The dissipative quantum phase transitions experienced by a driven optical system are studied in order to understand the underlying light-matter coupling. The primary system under study is composed of a cavity mode coupled to a two-level system with dissipation being introduced through the interaction with a surrounding environment; an external coherent field is included to drive the system out of the ground state. The coupling is given by a generalized Rabi Hamiltonian, where rotating, gr, and counter-rotating, ngr, couplings are both present and can be adjusted independently. A comprehensive study of the different phases the system can exhibit as gr and n are varied is presented. From this, we construct a bridge between two limiting scenarios: (i) the Jaynes-Cummings limit (n = 0), where the system undergoes a phase transition by means of the breakdown of the photon blockade, and (ii) the driven Dicke limit ( = 1 ), where the normal to super-radiant phase transition is found. Novel behaviour encountered in an intermediate regime (1 > n > 0) is discussed. Attention is drawn towards the strong coupling regime, where changes at the one-photon level induce nonlinear effects and behaviour reminiscent of phase transitions is encountered with just a few photons present. A comparison between weak and strong coupling regimes, and, thus, an exploration of the effect of fluctuations over quantum phase transitions of light, is given through a survey of the phases an auxiliary optical system exhibits. This system is composed of two coupled nonlinear cavities that are driven coherently and damped through the interaction with the environment; it is seen to exhibit three phases. Two phases present high-correlation between the cavities and are differentiated by the photon statistics, transitioning from classical to quantum. The third phase is characterized by a highly localized field in one of the cavities. The departure from mean-field results is highlighted and a new way to characterize the possible phases of the system is proposed. Finally, the crucial role of quantum fluctuations is quantified and used to define the phases at hand.
We show that the counter-rotating terms of the dispersive qubit-cavity Rabi model can produce relatively large and nonmonotonic Bloch-Siegert shifts in the cavity frequency as the system is driven through a quantum-to-classical transition. Using a weak microwave probe tone, we demonstrate experimentally this effect by monitoring the resonance frequency of a microwave cavity coupled to a transmon and driven by a microwave field with varying power. In the weakly driven regime (quantum phase), the Bloch-Siegert shift appears as a small constant frequency shift, while for a strong drive (classical phase) it presents an oscillatory behavior as a function of the number of photons in the cavity. The experimental results are in agreement with numerical simulations based on the quasienergy spectrum.
Out-of-equilibrium driven nonlinear oscillators have attracted interest as a platform for tailoring Floquet Hamiltonian spectra and stabilizing quantum states for quantum information. However, it is usually ignored that the necessary parametric drive can amplify environmental fluctuations limiting applications. In this work, we capture this effect by computing the effective Lindbladian of a Kerr parametric oscillator beyond the rotating wave approximation. We find that the relevant dissipators depend strongly on the nonlinearity and the mean-photon number in the oscillator. In particular, we address recent measurements of the logical lifetime on a Schr\"odinger's-cat qubit stabilized by a parametric drive, thereby providing an explanation for the observed three-order-of-magnitude discrepancy between experimental data and an ordinary Lindblad treatment. Our systematic approach paves the path to treat dissipative effects in Floquet-engineered nonlinear quantum systems.
Superconducting circuits embedding Josephson junctions leverage microwave drives for control and measurement of quantum states. Although increasing the drive power is desirable for improving the efficiency of these operations, it eventually triggers unwanted transitions to uncontrolled states. While careful choice of circuit symmetries and parameters can mitigate these effects, the presence of spurious circuit modes spoils the resilience to high power. In this work, we engineer a transmon-resonator system free of any detrimental unwanted transitions. This resilience enables us to access drive powers at which we uncover a remarkable physical phenomenon: the collapse and inversion of the Josephson potential. Through spectroscopy and readout experiments, we confirm that the driven potential goes to zero and inverts as the power increases. The inversion corresponds to the dynamical stabilization of the transmon at its unstable equilibrium point, directly analogous to an inverted pendulum. This result reveals a new limitation of strongly driven superconducting circuits beyond drive-induced transitions. In addition, the dynamical renormalization of the Josephson potential opens up novel avenues for the control of superconducting circuits, and the autonomous stabilization of noise-resilient quantum states.